NC functions over the nc Grassmannian

Abstract

We explain how to formulate Voiculescu's non-commutative Riemann sphere framework for fully matricial functions \cite{v10} within the theory of nc functions developed by Vinnikov and Kaliuzhnyi-Verbovetskyi \cite{kvv14}. We then extend this framework from the Riemann sphere to Grassmannians (and flag manifolds). Moreover, as an example of nc functions in this setting, we introduce a generalization of Voiculescu's non-commutative resolvent on the Riemann sphere, study a corresponding generalization of the resolvent equation, and discuss aspects of the spectral analysis of unbounded operators in Voiculescu's framework \cite{v10}.